Replacing GIS Payment Tables with Equations


The Canadian Guaranteed Income Supplement (GIS) payments for low income taxpayers is not easy to calculate because the oft-used 50% reduction from the yearly maximum for every dollar earned no longer holds for all income levels. The only data the Government of Canada provides are a number of tables that gives the GIS payment versus income ranges. To calculate the GIS payments in a spreadsheet and in applications, what is really required a set of equations.While working on the spreadsheet included in Estimating Savings Required for Retirement, issued in 2018, I came across an unexpected GIS result: the payments when close to the Maximum Annual Income Allowed were nowhere close to the small amount expected and instead where in the thousands of dollars. I used the well-known reduction of $1 in GIS annual payment for every $2 in annual income, but it does not match that in the tables the government provides.

After some research on the internet, I found a good description in a very argumentative thread on reddit.com from 2017 of what causes this discrepancy. One responder (nickname of flashstorm) indicates it is due to top-up amounts set in the 2011 and 2016 budgets that add to the amount specified in the OAS Act and links were provided to each of these items.

After reviewing them and using other sources, I have created a number of equations that can be used instead of the tables. I have adjusted the spreadsheet, as well as the calculation in Financebase-Lifetime Finances, to use the correct amounts. This post provides the logic used to create the equations. If you are not interested in the details, you can stop after the summary below. However, you might still find the background that follows useful before getting to the actual equations.

Summary

When you retire, you can receive the Old Age Security (OAS) pension and if your taxable income is low enough you may be eligible for the Guaranteed Income Supplement (GIS). Taxable income used in the calculation excludes the OAS and GIS payments as well as TFSA interest, but includes, for example, the Canada Pension Plan (CPP) payments, Registered Retirement Income Fund (RRIF) payments, pensions, dividends, and other interest payments.

The Canadian government calculates what your GIS payment, if any, for a year be based on your taxable income in the previous year. But you must apply for it, but only once as it is then done each year after that. They provide information on the payments provided on this page for the latest quarter: Old Age Security payment amounts. Scroll down to see the table that is for your situation and then select your income range.

While it is not obvious from the tables, they do not adhere to the well-know and often quoted reduction of the GIS of $1 for every $2 of income up to the maximum permitted per year before the GIS is no longer paid. This relationship was valid up to 2011 when the government provided a top-up in the 2011 budget – and then did it again in the 2016 budget.

The chart below shows the payments for a Single pensioner receiving a full OAS pension using the Oct to Dec 2016 numbers when both top-ups were available. The Basic GIS dashed red line is for the $1:$2 relationship. The solid blue line shows how the GIS has been increased for very low income taxpayers. The purple and green lines show the 2011 and 2016 budget amounts, respectively. The black line is the sum of the Income, OAS and GIS for a full OAS. The orange line shows just the Full OAS plus the GIS and decreases as income increases. There is a significant boost of income for very low income amounts.

The guaranteed part of the Guaranteed Income Supplement is included in the OAS Act. This provision ensures that even if a taxpayer does not qualify for a full OAS pension (40 years in Canada), the person will have the GIS amount increased so that the total OAS plus GIS is the same as for a person receiving a full OAS pension.

For example, every taxpayer who is single that qualifies for, and has applied for, a GIS would have received the amount shown on the orange line on the above chart which is the sum of the full GIS and OAS. For the Oct to Dec 2016 values, this means that a person who is single with zero taxable income will receive a total of $17,314.44, no matter what the OAS is. When the GIS goes to 0 at $17,544, the OAS will continue. The black line is the income plus the full GIS and OAS.

However, if a person is receiving a partial OAS pension, what happens when the Maximum Annual Income Allowed (of $17,544 for Oct-Dec 2016) is exceeded? We know that the partial OAS continues but there will still be some GIS payment because of the guarantee to provide the same combined GIS and OAS as if it were a full OAS pension.

For example, the full OAS payment in 2016 was $6,942.36. Using the case of a 60% partial OAS pension of $4,165.42 means that the GIS Payment at the maximum annual income allowed is for the remaining 40% or $2,776.94. If the GIS were to stop at the maximum income this would be a significant loss of income.

If appears that the government permits the GIS to continue after the maximum and reduces it by $1 for every $2 of income until it is gone. For this 60% example, this means that the GIS will continue for another $5,553.89 (= $2,776.94 * 2) or up to $23,097.89.

There does not appear to be any documentation available from the government to support this, so I have used the very good explanation provided by D. Runchey in How receiving a partial OAS pension affects GIS amounts.

The equations and algorithms created later in this post take both partial and full pensions into consideration. The following is the background for how this all came about.

OAS and GIS Payment Amounts

The Guaranteed Income Supplement (GIS) is part of the Old Age Security (OAS) program. You cannot receive GIS until you receive an OAS pension and you cannot receive OAS unless you have lived in Canada for at least 10 years (from age 18) and be 65 or older.

The amount of OAS payment is easy to calculate as it is the number of years in Canada, divided by 40, times the maximum for the year. Once you start to receive the OAS, it increases each year by an amount based on the Consumer Price Index (aka inflation). The OAS pension is “clawed-back” starting at an income in the low $70K’s at the rate of 15% of the income above the claw-back income and ends in the high $120K’s.

However, the GIS is only available if you are a low-income taxpayer with income below a specified taxable income. It is adjusted depending on how much other income you have and changes each 3 months to adjust for inflation.

The latest amounts for the OAS and GIS are given in this link: Old Age Security payment amounts. For October to December 2018, here are the amounts, which are used in the Lifetime Finances application for 2019. (There is also an Allowance, but this is not included as it is seldom claimed.)

Your situationMaximum monthly payment amountMaximum annual income to receive the OAS pension
Old Age Security (OAS) pension
Regardless of your marital status$600.85$123,386 (individual income)
Guaranteed Income Supplement (GIS) amounts for individuals receiving a full Old Age Security (OAS) pension.
If you are a single, widowed or divorced pensioner$897.42$18,216 (individual income)
If your spouse/common-law partner receives the full OAS pension$540.23$24,048 (combined income)
If your spouse/common-law partner does not receive an OAS pension$897.42$43,680 (combined income)

Using the above monthly amounts for a single pensioner, the yearly amounts of a full pension is OAS = $7,210.20 and GIS = $10,769.04. Notice that spouses/common-law partners that both receive a full OAS pension receive $121.24 less per month (2 x 540.23 = 1,080.46) than 2 singles (2 x 600.85 = 1,201.70) and that the maximum annual income allowed is only 32% more. (The government seems to subscribe to the view that it is cheaper to live together that separately, but it certainly does result in a reduction if you are married.)

If you scroll down on the government’s page, you can use Tables 1, 2 and 3 to view what you will receive for each of the above situations, respectively. You can also download a pdf version of all the tables. Here is an example of one of the 51 income ranges given in the drop-down menu for Table 1 for a Single Pensioner. Notice that for each income range, the GIS amount deceases by $1 per month, but the income range is not always $24 due to the top-ups.

Yearly Income (excluding OAS Pension and GIS)Monthly GIS with Maximum OAS PensionCombined Monthly OAS Pension and GIS
$1,968.00 – $1,991.99$815.42$1,416.27
$1,992.00 – $2,015.99$814.42$1,415.27
$2,016.00 – $2,039.99$813.42$1,414.27
2,040.00 – $2,047.99$812.42$1,413.27
$2,048.00 – $2,063.99$811.42$1,412.27
$2,064.00 – $2,087.99$810.42$1,411.27
$2,088.00 – $2,095.99$809.42$1,410.27

OAS Act and GIS Basic Payments

The Old Age Security Act of 1977 (with amendments) includes both the OAS and GIS payments. The values given in the table above are increased each 3 months by the Consumer Price Index resulting in the 2018 numbers given in the table above.
Here is what is in the Act for the OAS payment:
Basic amount of full pension

  • 7 (1) The amount of the full monthly pension that may be paid to any person for a month in the payment quarter commencing on January 1, 1985 is two hundred and seventy-three dollars and eighty cents.

The Act has the following for the GIS payment:
Amounts on April 1, 2005

  • 12 (1) The amount of the supplement that may be paid to a pensioner for any month in the payment quarter commencing on April 1, 2005 is,
  • (a) in the case of a person other than a person described in paragraph (b), five hundred and sixty-two dollars and ninety-three cents, and
  • (b) in the case of a person who, on the day immediately before that payment quarter, had a spouse or common-law partner to whom a pension may be paid for any month in that payment quarter,
    • (i) in respect of any month in that payment quarter before the first month for which a pension may be paid to the spouse or common-law partner, five hundred and sixty-two dollars and ninety-three cents, and
    • (ii) in respect of any month in that payment quarter commencing with the first month for which a pension may be paid to the spouse or common-law partner, three hundred and sixty-six dollars and sixty-seven cents,
  • minus one dollar for each full two dollars of the pensioner’s monthly base income.

Increase on January 1, 2006

  • (1.1) The amount of the supplement that may be paid to a pensioner for any month in the payment quarter commencing on January 1, 2006 is the amount of the supplement that would otherwise be payable plus
    • (a) eighteen dollars, in the case of a person described in paragraph (1)(a) or subparagraph (1)(b)(i); and
    • (b) fourteen dollars and fifty cents, in the case of a person described in subparagraph (1)(b)(ii).

Increase on January 1, 2007

  • (1.2) The amount of the supplement that may be paid to a pensioner for any month in the payment quarter commencing on January 1, 2007 is the amount of the supplement that would otherwise be payable plus
      • (a) eighteen dollars, in the case of a person described in paragraph (1)(a) or subparagraph (1)(b)(i); and
      • (b) fourteen dollars and fifty cents, in the case of a person described subparagraph (1)(b)(ii).

The result of the above is as follows:

The OAS monthly payment as of January 1, 1985 is $273.80 ($3,285.60/year).

For a Single pensioner, GIS monthly payment as of January 1, 2007 is $562.93 plus $18 plus $18 for a total of $598.93 ($7,187.16/year).

These are increased each year by the Consumer Price Index. I have used the Open Data Portal provided by the Government of Canada to download and then manipulate the Excel file downloaded from a search that resulted in this link. The following is a chart of the change in OAS and GIS over the past 30 years using this file.

The Guarantee in the GIS

The OAS and GIS payments are linked by a formula in section 12(5) that sets the

Guaranteed minimum income for pensioners

  • (5) Despite subsection (2), the amount of the supplement that may be paid to a pensioner for any month after December 1997 is the amount determined by the formula
    • [(A – B) × C] – D/2
    • where A is the aggregate of
      • (a) the maximum amount of the supplement that, but for this subsection, might have been paid to the pensioner for that month, and
      • b) the amount of the full monthly pension;
  • B is the pensioner’s monthly pension;
  • C is the pensioner’s special qualifying factor for the month; and
  • D is the pensioner’s monthly base income rounded, where it is not a multiple of two dollars, to the next lower multiple of two dollars.

This provision comes into play when the person does not receive a full OAS pension. The intent of this provision is to ensure that no matter what OAS payment is made, the GIS supplement plus maximum OAS in Tables 1 to 3 are paid to pensioners. The OAS then continues after the maximum income level given in the tables.

The A part of the formula is the GIS (the (a) subsection) plus the full OAS (the (b) subsection) as shown in the table above. The B part of the formula is the actual OAS payment received which may be less than the maximum due to less than 40 years of residence in Canada. The C part is not specified and seems to have a value of 1. The D/2 part does not seem to apply any more due to the two budget adjustments discussed below. Instead, I expect that Tables 1 to 3 on the government website are to be used instead.

Budget Adjustments to the GIS

There were increased GIS payments in two federal budgets. (The OAS Act was not changed.) The 2011 budget includes the following:

  • Effective July 1, 2011, seniors with little or no income other than Old Age Security and the Guaranteed Income Supplement will receive additional annual benefits of up to $600 for single seniors and $840 for couples. Single recipients with an annual income (other than Old Age Security and the Guaranteed Income Supplement) of $2,000 or less, and couples with an annual income of $4,000 or less, will receive the full amount of the benefit. Above these income thresholds, the amount of the top-up will be gradually reduced and will be completely phased out at an income level of $4,400 for singles and $7,360 for couples.

The 2016 budget further increases the top-up, but for singles only:

  • Single seniors with annual income (other than Old Age Security and Guaranteed Income Supplement benefits) of about $4,600 or less will receive the full increase of $947. Above this income threshold, the amount of the increased benefit will be gradually reduced and will be completely phased out at an income level of about $8,400. Benefits will be adjusted quarterly with increases in the cost of living.

As shown below, the above top-up annual payments, for a single person, are reduced at the rate of $1 for every $4 of yearly taxable income above the minimum amount indicated until the phase-out amount is reached when the payment becomes zero.

Replacing the Tables with Equations

What I really want is an algorithm or a set of equations that I can use in my spreadsheets and applications to determine the payments without using Tables 1, 2 and 3 which change every 3 months. What works is to use straight lines for the different income ranges. The equation used is: y = mx + b, where y is the GIS payment, x is the income, m is the slope and b (the y-intercept) is the GIS payment when income is zero. The objective is to obtain values for the y-intercept (b).

I have used the Oct to Dec 2016 OAS and GIS amounts as these are for the end of the year after the last top-up. To obtain the data for this period, I used the Government of Canada Publications site using this link to get access to the archived pdf files for the 2013 to 2018 “Table of benefit amounts by marital status and income level : Old Age Security Pension, Guaranteed Income Supplement (GIS)”. While the data in the file are by month, all equations below are for a year.

Equations for a Single Pensioner

The payments are given in Table 1 on the government website and the pdf file mentioned above. The 2011 budget and 2016 budget apply as well as the Basic payment as discussed below.

An equation for the 2011 budget payments is not difficult to create. We know from the budget that it was $600 per year up to $2,000 and then decreased to end at about $4,400. To make the payment go to zero at about $4,400, a slope is needed. Using a slope of -1/4, results in the 2011 budget payment going to zero at $2,000 + (4 * $600) = $4,400 which is exactly what the government said it would be.

In October 2016, the $600 amount was increased by inflation to $650.40 according to the same file used for the inflation information given above. While not indicated anywhere, income when the reduction starts has to be increased by the same inflation amount to $2,028 for the equations to compare well to the actual GIS amounts. Therefore, the equations are as follows:

  • 2011 Budget Yearly GIS Payment
  •       = $650.40, for yearly income up to $2,028.
  •       = (-1/4) * (Yearly Income – 2,028) + $650.40, from $2,028 until it is gone.

Because the payment has increased, the income when the top-up goes to zero has also increased and is expressed by this equation:

  • 2011 Budget Income when GIS is $0
  •       = (Change in Income per $1 GIS * GIS at $0 Income) + Income when Reduction Starts 
  •       = (4 * $650.40) + $2,028
  •       = $4,629.60

A similar equation for the 2016 budget payments can be created. We know from the budget statement above that it was $947 per year up to $4,600 and then decreased to end at about $8,400.

To make the payment go to zero at about $8,400, a slope is needed. Given that the 2011 budget GIS slope is -1/4, it is only natural to assume that the government would choose the same number for the 2016 budget value. Using a slope of -1/4, results in the 2016 budget payment going to zero at $4,600 + (4 * 947) = $8,388. This is close enough to the amount in the budget.

In October 2016, the $947 amount increased by inflation to $955.56 according to the same file used for the inflation information given above.

To make the straight lines fit the actual GIS values, I made a decision that the income when the reduction starts will be equal to the income when the 2011 budget amount stops. This has been calculated using the equation above. Therefore, the equations are as follows:

  • 2016 Budget Yearly GIS Payment
  •       = $955.56, for yearly income up to $4,629.60.
  •       = (-1/4) * (Yearly Income – 4,600) + $955.56, from $4,629.60 until it is gone.

For the Basic payment the slope (m) is -1/2 which is the reduction of $1 per month for every $2 of yearly income as per the OAS Act. The GIS at $0 Income is for whatever has not been assigned to the two top-ups given above and decreases by the slope. The starting value is as follows:

  • Basic GIS Payment at $0 Income
  •       = Total GIS Payment at $0 Income – 2011 Budget GIS Payment at $0 Income – 2016 Budget GIS Payment at $0 Income
  •       = $10,369.08 – $650.40 – $955.56
  •       = $8,763.12

The equation to be used is as follows:

  • Basic Yearly GIS Payment
  •       = (-1/2) * Yearly Income + $8,763.12

Equations for Couple, Both Receiving OAS

The payments are given in Table 2 on the government website and the pdf file mentioned above. Only the 2011 budget applies and in 2011 it had an amount of $840 (for the couple) up to a combined income of $4,400 and then decreases to zero at the phase-out income of $7,360. These numbers are adjusted below for the October 2016 amounts.

An equation for the 2011 budget payments is created in the same way as for a single pensioner. We know from the budget that it was a total of $840 per year for both people up to $4,000 and then decreased to end at about $7,360. Using a slope of -1/4, results in the 2011 budget payment going to zero at $4,000 + (4 * $840) = $7,360 which is exactly what the government said it would be.

The equations are for only 1 person, so that in Oct 2016, the original amount of $840 per couple or $420 per person has increased by inflation to $455.28 per person according to the same file used for the inflation information given above. While not indicated anywhere, income when the reduction starts has to be increased by the same inflation amount to $4,056 for the equations to compare well to the actual GIS amounts. In addition, the slope has to be changed by a factor of 2 from -1/4 to -1/8 because there are 2 people splitting the GIS. Therefore, the equations are as follows, for each person:

  • 2011 Budget Yearly GIS Payment
  •       = $455.28, for yearly income up to $4,056.
  •       = (-1/8) * (Yearly Income – 4,056) + $455.28, from $4,056 until it is gone.

For the Basic payment the slope (m) is -1/4 (because there are 2 people) which is the reduction in the GIS of $1 for every $4 of income. The GIS at $0 Income is for whatever has not been assigned to the 2011 top-up given above and decreases by the slope. The starting value is as follows:

  • Basic GIS Payment at $0 Income
  •       = Total GIS Payment at $0 Income – 2011 Budget GIS Payment at $0 Income
  •       = $6,242.04 – $455.28
  •       = $5,786.76

The equation to be used is as follows:

  • Basic Yearly GIS Payment
  •       = (-1/4) * Yearly Income + $5,786.76

Equations for Couple, One Receives OAS

Getting the equations for a couple with only one person receiving the OAS (normally this means that the other person is still working) is not obvious. The 2011 budget amounts apply, but because it is a couple, the 2016 budget probably should not apply. However, it does, probably because the person is “single” for the purposes of the GIS. However, the tables have a much larger Maximum Annual Income Allowed.

Examining Table 3 for Oct 2016, the income ranges given do not start at $0 income before a reduction, but at $4,096. There is no indication anywhere that I can find as to why this happens. During the $0 to $4,095.99 initial period the GIS is the same maximum as for a single and then only drops until it is all gone at the $42,048 Maximum Annual Income Allowed.

To determine what equations are to be used, the complete Table 3 GIS amount was charted against the From income value. It showed the line changing slope as below:

  • From $0 to $4,096 it is a straight line.
  • It has a small slope from there to about $6,900.
  • The slope increases from there to about $16,800.
  • After that the slope is 1:4 to the end.

The equations are created starting with the two budget amounts (the same as for a single but with different slopes because the maximum annual income allowed is much greater) and using what is left as the Basic amount. The following equations are the result of getting them to fit the chart of actual values for Oct 2016:

  • 2011 Budget Yearly GIS Payment
  •       = $650.40, for yearly income up to $4,095.99.
  •       = (-1/8) * (Yearly Income – 4,095.99) + $650.40 – $12, from $4,096 until it is gone. The $12 is to ensure that the first amount after 4,095.99 is reduced properly.

For the 2016 budget here are the equations:

  • 2016 Budget Yearly GIS Payment
  •       = $955.56, for yearly income up to the income when the 2011 budget amount is all gone.
  •       = (-1/8) * (Yearly Income – 9,299.19) + $955.56, after the 2011 budget amount is all gone.

To determine when the 2011 Budget amount is all gone use the following equation:

  • 2011 Budget Income when GIS is $0
  •       = 8 * 2011 Budget Yearly GIS Payment before the reduction  + 4,095.99
  •       = 8 * $650.40 + $4,095.99
  •       = $9,299.19

The Basic equation is also flat for some income before it starts its slope to the end. The income where the reduction starts is calculated so that the GIS value is whatever is left-over from the full amount less the two budget values.

  • Basic Yearly GIS Payment when Income is $0
  •       = Total GIS at $0 Income – 2011 Budget Yearly GIS Payment when Income is $0 – 2016 Budget Yearly GIS Payment when Income is $0
  •       = $10,369 – $650.40 – $955.56
  •       = $8,763.12

The income at which the Basic GIS starts its downward slope is obtained from the following equation:

  • Income when Basic reduction starts
  •       = Income at last table entry – (4 * (Basic Yearly GIS Payment when Income is $0 – (12 * GIS at last table entry)))
  •       = $42,000 – (4 * ($8,763.12 – (12 * $.26))) = $42,000 – (4 * $8760)
  •       = $42,000 – $35,040
  •       = $6,960

This results in the equations for the Basic amount:

  • Basic Yearly GIS Payment
  •       = $8,763.12, for yearly income up to $6,960.
  •       = (-1/4) * (Yearly Income – $6,960) + $650.40, from $6,960 until it is gone.

Partial Pension

The GIS payment for a pensioner is adjusted from the amounts in the tables if the person has less than the maximum OAS payment. The formula given in the section in the OAS Act called “Guaranteed minimum income for pensioners” discussed above ensures that the sum of the GIS and OAS payments are the same for all single pensioners, even if a partial OAS pension is paid. As a result, two things happen.

1) The GIS is increased at any income level to match the reduction in the OAS pension. This means that the new GIS payment is as follows:

  • New Yearly GIS Payment
  •       = Calculated Yearly GIS Payment + (Full Yearly OAS Payment – Actual Yearly OAS Payment Received), for all income up to the Maximum Annual Income Allowed

One advantage to the taxpayer of this arrangement is that the increased GIS payment is not subject to income taxes while the OAS is always taxed.

2) When the income exceeds the Maximum Annual Income Allowed for GIS payments, the partial OAS payment continues and the remaining GIS amount is reduced by the ratio for the Basic yearly payment: 1:2 for a Single or 1:4 for a couple. The maximum income is adjusted as follows:

  • Adjusted Maximum Annual Income Allowed
  •       = Maximum Annual Income Allowed + (Ratio * (Full Yearly OAS Pension – Actual Yearly OAS Pension)) Received)), where Ratio is the Change in Income for $1 GIS is 2 for a Single and 4 for a couple.

During the income range between the Maximum Annual Income Allowed and the Adjusted Maximum Annual Income Allowed the GIS payment is as follows:

  • Yearly GIS Payment between Maximum and Adjusted
  •       = (Full Yearly OAS Pension – Actual Yearly OAS Pension) + ((-1/Ratio) * (Income – Maximum Annual Income Allowed)), where Ratio is the Change in Income for $1 GIS is 2 for a Single and 4 for a couple.

This guaranteed minimum income must be taken into account when applying the equations created above. The simplest way to do this is to use the equations for the full OAS pension as above and then just subtract the actual OAS to get the GIS. However, this means extra calculations (or columns on spreadsheets) are needed for the full OAS+GIS.

Changes Needed for Future Years

The above analysis used the Oct – Dec 2016 data because it was the first year that had both budget top-ups. However, we are in 2019 and the GIS/OAS tables have changed. I created spreadsheets for 2017 and 2018 to check if the equations provided the proper results when inflation is included. I also had to update the History spreadsheet to get the 2011 and 2016 budget amounts.

The results were that the inflated values for the OAS and GIS amounts only give the correct results when using the equations if the Income when the reduction starts for the 2011 budget is not inflated. That is, they must stay at $2,028 for a single and $4,056 for a couple. (Recall that I decided the reduction for the 2016 budget starts when the 2011 budget amount goes to 0.)

When using these equations in the spreadsheets and Lifetime Finances, the Oct-Dec values for the year prior to the current year must be used because inflation is applied to the amounts at the end of the year values. For example, for Lifetime Finances, the Oct-Dec 2018 GIS values for the 2019 year are as follows:

SituationTotal BenefitMax Income2011 benefit2011 reduction2016 benefitBasic reduction
Single10,769.0418,216.00675.482,028.00992.520.00
Married, both with OAS6,482.7624,048.00472.684,056.000.000.00
Married, one with OAS10,769.0443,680.00675.484095.99992.527,248.00

On the spreadsheets and in Lifetime Finances, the benefits and the Maximum Annual Income Allowed are increased by the appropriate inflation rate at the end of the year.

Final Comments

The GIS is a complicated program and actual amounts are only accurate when done by the federal agency involved. For example, amounts are calculated using the past year’s income tax return(s) and do not take effect until part way into the next year. Therefore, only use the results given in the spreadsheets and Lifetime Finances as an approximation as the best that can be done using the equations developed above.

If you are interested in the spreadsheets that I used to verify the equations, just send me a note using the form on the Contact page and I can send them by email.